Since any Riemannian manifold, satisfying only mild topological finiteness conditions (with or without boundary) admits a fat triangulation – see e.g. [1]} and the references therein, it follows that any \(\varepsilon \) -net can be improved to render a thick triangulation.
Furthermore, during the thickening process the edge lengths (weights) are modified only slightly, given that one uses to this end only \(\varepsilon \) -moves (see [2]}, [3]} for a technical details).
Given the facts above an recalling that, by Formula (REF ), the full Forman-Ricci curvature is expressed by
\(\hspace*{-75.39963pt}{\rm Ric}_{\rm F} (e) = \omega (e) \left[ \left( \sum _{e \sim f} \frac{\omega (e)}{\omega (f)}+\sum _{v \sim e} \frac{\omega (v)}{\omega (e)} \right) \right.\) 
\(\hspace*{75.39963pt}- \left. \sum _{\hat{e} \parallel e} \left| \sum _{\hat{e},e \sim f} \frac{\sqrt{\omega (e) \cdot \omega (\hat{e})}}{\omega (f)} - \sum _{v \sim e, v \sim \hat{e}} \frac{\omega (v)}{\sqrt{\omega (e) \cdot \omega (\hat{e})}} \right| \right] \; ;\) 
